اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدResonance Frequencies of a Slab with Subwavelength Slits: a Fourier-transformation Approach
83
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
This paper proposes a novel, rigorous and simple Fourier-transformation approach to study resonances in a perfectly conducting slab with finite number of subwavelength slits of width $hll 1$. Since regions outside the slits are variable separated, by Fourier transforming the governing equation, we could express field in the outer regions in terms of field derivatives on the aperture. Next, in each slit where variable separation is still available, wave field could be expressed as a Fourier series in terms of a countable basis functions with unknown Fourier coefficients. Finally, by matching field on the aperture, we establish a linear system of infinite number of equations governing the countable Fourier coefficients. By carefully asymptotic analysis of each entry of the coefficient matrix, we rigorously show that, by removing only a finite number of rows and columns, the resulting principle sub-matrix is diagonally dominant so that the infinite dimensional linear system can be reduced to a finite dimensional linear system. Resonance frequencies are exactly those frequencies making the linear system rank-deficient. This in turn provides a simple, asymptotic formula describing resonance frequencies with accuracy ${cal O}(h^3log h)$. We emphasize that such a formula is more accurate than all existing results and is the first accurate result especially for slits of number more than two to our best knowledge. Moreover, this asymptotic formula rigorously confirms a fact that the imaginary part of resonance frequencies is always ${cal O}(h)$ no matter how we place the slits as long as they are spaced by distances independent of width $h$.
قيم البحث
اقرأ أيضاً
This paper presents a simple Fourier-matching method to rigorously study resonance frequencies of a sound-hard slab with a finite number of arbitrarily shaped cylindrical holes of diameter ${cal O}(h)$ for $hll1$. Outside the holes, a sound field can
be expressed in terms of its normal derivatives on the apertures of holes. Inside each hole, since the vertical variable can be separated, the field can be expressed in terms of a countable set of Fourier basis functions. Matching the field on each aperture yields a linear system of countable equations in terms of a countable set of unknown Fourier coefficients. The linear system can be reduced to a finite-dimensional linear system based on the invertibility of its principal submatrix, which is proved by the well-posedness of a closely related boundary value problem for each hole in the limiting case $hto 0$, so that only the leading Fourier coefficient of each hole is preserved in the finite-dimensional system. The resonance frequencies are those making the resulting finite-dimensional linear system rank deficient. By regular asymptotic analysis for $h ll 1$, we get a systematic asymptotic formula for characterizing the resonance frequencies by the 3D subwavelength structure. The formula reveals an important fact that when all holes are of the same shape, the Q-factor for any resonance frequency asymptotically behaves as ${cal O}(h^{-2})$ for $hll1$ with its prefactor independent of shapes of holes.
In this paper a deterministic sparse Fourier transform algorithm is presented which breaks the quadratic-in-sparsity runtime bottleneck for a large class of periodic functions exhibiting structured frequency support. These functions include, e.g., th
e oft-considered set of block frequency sparse functions of the form $$f(x) = sum^{n}_{j=1} sum^{B-1}_{k=0} c_{omega_j + k} e^{i(omega_j + k)x},~~{ omega_1, dots, omega_n } subset left(-leftlceil frac{N}{2}rightrceil, leftlfloor frac{N}{2}rightrfloorright]capmathbb{Z}$$ as a simple subclass. Theoretical error bounds in combination with numerical experiments demonstrate that the newly proposed algorithms are both fast and robust to noise. In particular, they outperform standard sparse Fourier transforms in the rapid recovery of block frequency sparse functions of the type above.
We consider the inverse problem of estimating the spatially varying pulse wave velocity in blood vessels in the brain from dynamic MRI data, as it appears in the recently proposed imaging technique of Magnetic Resonance Advection Imaging (MRAI). The
problem is formulated as a linear operator equation with a noisy operator and solved using a conjugate gradient type approach. Numerical examples on experimental data show the usefulness and advantages of the developed algorithm in comparison to previously proposed methods.
Nonequispaced discrete Fourier transformation (NDFT) is widely applied in all aspects of computational science and engineering. The computational efficiency and accuracy of NDFT has always been a critical issue in hindering its comprehensive applicat
ions both in intensive and in extensive aspects of scientific computing. In our previous work (2018, S.-C. Yang et al., Appl. Comput. Harmon. Anal. 44, 273), a CUNFFT method was proposed and it shown outstanding performance in handling NDFT at intermediate scale based on CUDA (Compute Unified Device Architecture) technology. In the current work, we further improved the computational efficiency of the CUNTTF method using an efficient MPI-CUDA hybrid parallelization (HP) scheme of NFFT to achieve a cutting-edge treatment of NDFT at super extended scale. Within this HP-NFFT method, the spatial domain of NDFT is decomposed into several parts according to the accumulative feature of NDFT and the detailed number of CPU and GPU nodes. These decomposed NDFT subcells are independently calculated on different CPU nodes using a MPI process-level parallelization mode, and on different GPU nodes using a CUDA threadlevel parallelization mode and CUNFFT algorithm. A massive benchmarking of the HP-NFFT method indicates that this method exhibit a dramatic improvement in computational efficiency for handling NDFT at super extended scale without loss of computational precision. Furthermore, the HP-NFFT method is validated via the calculation of Madelung constant of fluorite crystal structure, and thereafter verified that this method is robust for the calculation of electrostatic interactions between charged ions in molecular dynamics simulation systems.
We investigate the use of spatial interpolation methods for reconstructing the horizontal near-surface wind field given a sparse set of measurements. In particular, random Fourier features is compared to a set of benchmark methods including Kriging a
nd Inverse distance weighting. Random Fourier features is a linear model $beta(pmb x) = sum_{k=1}^K beta_k e^{iomega_k pmb x}$ approximating the velocity field, with frequencies $omega_k$ randomly sampled and amplitudes $beta_k$ trained to minimize a loss function. We include a physically motivated divergence penalty term $| abla cdot beta(pmb x)|^2$, as well as a penalty on the Sobolev norm. We derive a bound on the generalization error and derive a sampling density that minimizes the bound. Following (arXiv:2007.10683 [math.NA]), we devise an adaptive Metropolis-Hastings algorithm for sampling the frequencies of the optimal distribution. In our experiments, our random Fourier features model outperforms the benchmark models.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
